Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.453 Quantum Optical Communication Lecture Number 18 Fall 2016 Jeffrey H. Shapiro � c 2008, 2010, 2012, 2014, 2015 Date: Tuesday, November 15, 2016 Reading: For random processes: • J.H. Shapiro, Optical Propagation, Detection, and Communication , chapter 4. Reading: For continuous-time photodetection: • J.H. Shapiro, H.P. Yuen, and J.A. Machado Mata, “Optical communication with two-photon coherent states—Part II: photoemissive detection and structured receiver performance,” IEEE Trans. Inform. Theory IT-25 , 179–192 (1979). • H.P. Yuen and J.H. Shapiro, “Optical communication with two-photon coherent states—Part III: quantum measurements realizable with photoemissive detec- tors,” IEEE Trans. Inform. Theory IT-26 , 78–92 (1980). • J.H. Shapiro, “Quantum noise and excess noise in optical homodyne and het- erodyne receivers,” IEEE J. Quantum Electron. QE-21 , 237–250 (1985). • L. Mandel and E. Wolf Optical Coherence and Quantum Optics, (Cambridge University Press, Cambridge, 1995) sections 9.1–9.8, 12.1–12.4, 12.9, 12.10. • J. H. Shapiro, “The Quantum Theory of Optical Communications,” IEEE J. Sel. Top. Quantum Electron. 15 , 1547–1569 (2009); J.H. Shapiro, “Corrections to ‘The Quantum Theory of Optical Communications’,” IEEE J. Sel. Top. Quantum Electron. 16 , 698 (2010). Introduction Today we begin a two-lecture treatment of semiclassical versus quantum photode- tection theory in a continuous-time setting. We’ll build these theories in a slightly simplified framework, i.e., scalar fields 1 with no ( x, y ) dependence illuminating the active region of a photodetection that lies within a region A of area A in a constant- z 1 These scalar fields may be regarded as being linearly polarized (for the classical case) or only being excited in one linear polarization (for the quantum case). 1
plane. 2 Also, we’ll focus our attention on almost-ideal photodetection, i.e., we will allow for sub-unity quantum efficiency ( η < 1), but otherwise our detector will be the continuous-time version of the ideal photodetector that we treated earlier this semester within the single-mode construct. The particular tasks we have set for to- day’s lecture are to develop the semiclassical and quantum photodetection statistical models for direct detection, and to exhibit some continuous-time signatures of non- classical light. Because this work will require elements of random process theory, supplementary notes covering the background we shall assume have been provided. We will begin our treatment by reprising some remarks, from Lecture 8, about the re- lationship between real photodetectors and the idealized model that we shall employ today and next time. A Real Photodetector Slide 3 shows a theorist’s cartoon of a real photodetector. The two large blocks on this slide are the photodetector and the post-detection preamplifier. The smaller blocks within the two large blocks are phenomenological, i.e., they do not represent discrete components out of which the larger entities are constructed. Nevertheless, it is instructive to walk our way through this photodetection system by means of these phenomenological blocks. Incoming light—whether we model it in classical or quantum terms—illuminates an optical filter that models the wavelength dependence of the photodetector’s sensitivity. The light emerging from this filter then strikes the core of the photodetector, i.e., the block that converts light into a light-induced current, which we call the photocurrent. Photodetectors have some current flow in the absence of illumination, and this dark current adds to the photocurrent within the detector. High-sensitivity photodetectors—such as avalanche photodiodes and photomultiplier tubes—have internal mechanisms that amplify (multiply) the initial photocurrent (and the dark current), and we have shown that on Slide 3 as a current multiplication block. 3 This current multiplication in general has some randomness associated with it, imposing an excess noise on top of any noise already inherent in the photocurrent and dark current. The electrical filter that is next encountered models the electrical bandwidth of the photodetector’s output circuit, and the thermal noise generator models the noise associated with the dissipative elements in the detector. Because the output current from a photodetector may not be strong enough to re- gard all subsequent processing as noiseless, we have included the preamplifier block in Slide 3. Its filter, noise generator, and gain blocks model the bandwidth character- istics, noise figure, and gain of a real preamplifier. Ordinarily, the output from such 2 The absence of transverse dependence means that only the normally-incident plane-wave com- ponent of the electromagnetic field is non-zero (in the classical case) or excited (in the quantum case). 3 We have shown the photocurrent and dark current as undergoing the same multiplication pro- cess. In real detectors, these two currents may encounter different multiplication factors. 2
a preamplifier is strong enough that any further signal processing can be regarded as noise free. An Almost-Ideal Photodetector Because we are interested in the fundamental limits of photodetection—be they repre- sented in semiclassical or quantum terms—we will strip away almost all the non-ideal elements of the real photodetection system shown on Slide 3, and restrict our atten- tion to the almost-ideal photodetector shown on Slide 4. 4 Be warned, however, that in experimental work we cannot always ignore the phenomena cited in our discussion of Slide 3. Nevertheless, it does turn out that there are photodetection systems that can approach the following almost-ideal behavior under some circumstances. Our almost-ideal photodetector is a near-perfect version of the photocurrent gen- erator block from Slide 3. In particular, our almost-ideal photodetector has these properties. • Its optical sensitivity covers all frequencies. • Its conversion of light into current has efficiency η . • It does not have any dark current. • It does not have any current multiplication. • It has infinite electrical bandwidth. • Its subsequent preamplifier has infinite bandwidth and no noise, so it need not be considered as it does not degrade the photodetection performance. As a result, the photocurrent takes the form of a random train of area- q impulses, where q is the electron charge, and a counting circuit driven by this photocurrent will produce, as its output, a staircase function of unit-height steps which increments when each impulse occurs, i.e., the photocount record � t 1 N ( t ) = du i ( u ) , for 0 ≤ t ≤ T , (1) q 0 as shown on Slide 4. 4 The sole non-ideality that we shall include is the sub-unity quantum efficiency, η < 1, of the photocurrent generator. This non-ideality is relatively easy to incorporate into our analysis, as we have already seen for the single-mode case. Moreover, η < 1 has a major impact on the utility of non-classical light, as we have seen for the single-mode analysis of the squeezed-state waveguide tap. Thus it is important to retain its effects in our continuous-time treatment. 3
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